Is it possible for an argument to be valid by virtue of its logical form, but contain a false premise? In other words, can a premise be false even though the argument itself is logically valid?

Thanks in advance!

(For context: the initial question was whether an argument can be false even though it "seems true" in terms of its logical form. The question is better asked this way: can the propositions contained in an argument be false even though the argument itself is logically valid?)

@rus9384 - The OP asked for a valid argument with false premises, and this is one.
– Mauro ALLEGRANZAMay 8 '18 at 12:09

3

You can see Validity and Soundness to review the basic definitions : "Whether or not the premises of an argument are true depends on their specific content. However, according to the dominant understanding among logicians, the validity or invalidity of an argument is determined entirely by its logical form. The logical form of an argument is that which remains of it when one abstracts away from the specific content of the premises and the conclusion, leaving only those elements that are common to discourse and reasoning about any subject matter, ... 1/2
– Mauro ALLEGRANZAMay 8 '18 at 13:11

10 Answers
10

First: we don't really say that arguments are true or false. Statements are true or false, but arguments have different kinds of properties.

One of those properties is, as you are obviously aware of, validity. However, another important property is well-foundedness, which means that the premises are true (or, for more practical everyday purposes, plausible or acceptable).

Well-foundedness is important, because if I am allowed to just assume anything as my premise, I can (validly!) argue for anything. For example:

"All dogs are purple. Foofy is a dog. Therefore, Foofy is purple"

This argument is logically valid, but not well-founded. And indeed, as such it is a bad argument.

... which is probably just what you were looking for when you said you wanted a valid but 'false' argument. Indeed, instead of saying that arguments are true or false, you can say they are good or bad (and of course anything in between: pretty good, pretty bad, ho-hum, excellent, terrible, etc.)

A special kind of 'bad' argument is something like this:

"Bananas are yellow. Therefore, bananas are yellow"

Interestingly, this argument is logically valid, and its premises are true (well, not in my local supermarket, which for some reason thinks that I would like to purchase their still green bananas, but you get the point). However, it is what you will recognize as a circular argument ... which is bad. OK, but why exactly is it bad? Well, think about it: why would someone be looking for an argument as to whether bananas are yellow or not? Presumably it is exactly because such a person doesn't know whether bananas are yellow or not. And we really shouldn't be assuming something that, to this person, is not acceptable ... which is another reason why for real life purposes, it may be more useful to define well-foundedness as 'the premises are acceptable' rather than 'the premises are true'.

Thank you. So if an argument takes a logically valid form, it is still possible for the content in one of its propositions to be false?
– Curious May 8 '18 at 11:48

@Curious If an argument is valid, all that means is that if all of the premises were to be true, then the conclusion would necessarily be true as well, as in, it is not possible for the conclusion to be false. So, it is possible for an argument to be valid and have false propositions, it just needs to be the case that if the premises were true, then the conclusion would also be true. However, it is therefore not possible for a valid argument to have all true premises but a false conclusion, that is not valid.
– Not_HereMay 8 '18 at 12:12

3

@Curious correct. In fact, it is possible for all claims in a valid argument to be false. Example: "All dogs are purple. All purple objects are square. Therefore,, all dogs are square."
– Bram28May 8 '18 at 12:26

1

Green bananas are desirable when you don't intend to consume them within the next day or so.
– jpmc26May 9 '18 at 2:26

2

@Harabeck No. Well-foundedness refers merely to the premises being true/acceptable. It says nothing about validity, just as vality sasys nothing about well-foundedness. But soundness is both well-foundedness and validity: sound=well-founded+valid.
– Bram28May 10 '18 at 10:17

The argument is valid because there is a correct relation between premises and conclusion. This is not because the conclusion is actually true but, crucially, because granted the premises the conclusion must follow even though one of the premises is false.

As noted in the answer above [Bram28], an argument itself is never said to be true or false : truth and falsity belong only to the premises and the conclusion. Arguments are only valid or invalid, depending on how premises and conclusion are related. If we cannot affirm the premises and consistently deny the conclusion, the argument is valid.

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

The following is a valid argument:

All dogs hate cats

Fluffy is a dog

Therefore, Fluffy hates cats.

Its only a sound argument if both premises are true. I can tell you that they are not, since Fluffy is my hamster.

It sounds like you are trying to ask if you can have logical premises that are false, yet support a conclusion that is true - in other words, an example of presenting facts that lead to a true statement, but the facts themselves are wrong.

This is entirely possible - the other answers provided give absurd examples of demonstrably false things, but they still logically lead to the conclusion presented being true, and the argument is true.

Take @Bram28 's purple dog for example - assume that what actually happened is that someone has poured purple paint on a cat named Floofy, and the damp cat looks like a small dog. This means the conclusion (Floofy is Purple) true, but the premises (All dogs are purple/Floofy is a dog) false.

Yes, it's always interesting to note that false premises can validly lead to a true conclusion. +1 But I would refrain from calling the argument 'true' ... truth is a property of statements, not arguments. So in your edited example (thanks for picking mine BTW :) ), the conclusion is true, but the argument is valid.
– Bram28May 8 '18 at 17:27

@Bram28Thank you for the comment - I've edited the answer to refer to what I was calling an 'argument' as a 'conclusion'.
– ZibbobzMay 9 '18 at 13:48

If the premises are true and the logic is valid, then the conclusion must be true.

But if either the premises are false or the logic is invalid, then the conclusion does not follow.

For example:

All bald men are master criminals.

Fred is bald.

Fred is a criminal.

The logic is impeccable. If it is true that all members of group X have characteristic Y, and if X1 is a member of group X, than X1 must have characteristic Y. The flaw is that it is not true that all bald men are master criminals.

Example 2:

Lex Luthor is bald.

Lex Luthor is a master criminal.

Therefore all bald men are master criminals.

Here the premises are all true. Lex Luthor really is bald, and Lex Luthor really is a master criminal. But the logic is flawed. Just because one member of group X has characteristic Y doesn't mean that all members of group X have characteristic Y.

Of course sometimes a conclusion will be true even though the argument does not prove it. Maybe Fred really is a master criminal. But that's essentially a coincidence.

As well as soundness and validity and such, it may also be worth considering that an argument can (although it may be considered poor form) have redundancies. In that case, it could be that one false premise does not break the overall argument.

For example:

The clouds are black

The weatherman predicts rain

If the clouds are black or the weatherman predicts rain then it will rain.

It will rain.

Even if one of the first two presmises is false, the conclusion not only could be true but must still be true. (Of course if the third claim is false, that is not the case.)

Technically, in all classical, standard logics, if an argument with any number of true premises is valid, its conclusion must be true as well, based on the fixed meaning of the logical constants occurring in them. If you then add an arbitrary false premise to the given true premises, this does not invalidate the argument, as long as the conclusion remains the same and hence does not use anything from of the newly added false premise.

For this reason, some proof theories require that all premises are in fact used in the proof.

The idea of adding an arbitrary proposition or statement as a proof theory step is not that unusual, as the core of the logical rule of disjunction introduction in natural deduction systems allows exactly that. This proof rule tells you to disjoin at any point in a proof a proposition/statement with any arbitrary one. In practice you look at the conclusion to see what you need to make it to that end, and hence the arbitrariness becomes de facto less free and much more constrained.

I made an edit which you may roll back or continue editing. Basically I fixed spelling and reformatted into paragraphs. You may see the versions by clicking on the "edited" link above. You mentioned "some proof theories", would you have references for them? Welcome to this SE!
– Frank HubenySep 18 '18 at 22:25

The premise establishes the area being discussed.
If the premise is conditional to the validity of the logic of the argument, then it must be true for the logic to be valid.
There are an infinite number of premises that could be invented for anything
but only ones relevant to the logic need to be applied.

So for a logical argument to be valid all the assumptions and premises used
need to be quantified and be valid for the logic to hold true.
In practical terms people make mistakes, get the wrong premises, but
are still correct in their logic, because it is founded on valid premises
which are not specified. You could argue the argument is then invalidated
because the correct premises where not stated. So it depends on what is
actually being reviewed, the argument or the argument with its complete set
of premises.

I disagree with the majority of answers given, which all fall into the same logical trap. They all follow the form:

a is true for all X

Y is an X

therefore a is true for all Y

However, a can be true for all Y for other reasons. The conclusion might be valid, but the argument is not.

This becomes more obvious with a nonsensical statement:

The sky is blue

My dress is the sky

Therefore, my dress is blue

Obviously, a dress cannot be a sky (category error), so its blueness cannot depend on this non-fact. The fact that it is actually blue fits to the argument by coincident (or because it was crafted this way), but it does not actually follow from the argument.

You are confusing argument content with argument form. Deductive reasoning studies argument form which could apply to any subject or predicate universally. Valid in the mathematical sense is what most people hear and refer to these days. Mathematical logic expresses validity is all that matters in certain arguments.----not all arguments. Many arguments are not mathematical. This is why you responded the way you did. But your method only works because you understand the content of the premises. You would likely fail if you had premises you did not know the value of.
– LogikalMay 9 '18 at 22:45